Limits of OM, XRD and SEM

BTNN

BT

NN

EDS

G.B.

G.B.

BT-BaTiO3

NN-NaNbO3G.B.-Grain boundaryM.G.J.-multiple grain junction

•Lateral resolution: ~µm•Details of microstructure:e.g., domain structure,chemical inhomogeneityphase distribution, grainboundaries, interfaces,precipitates, dislocations,etc.

← NN/BT

M.G.J.0.2µm

2-D Reciprocal Lattices

Real space:Unit cell vectors: a,b

d-spacing directiona d10 [10]b d01 [01]

Reciprocal space:Unit cell vectors: a*,b*

magnitude directiona* 1/d10 ⊥ bb* 1/d01 ⊥ a

A reciprocal lattice can be built usingreciprocal vectors.

[01]

[10] (10)

(01)

Note: each point in the reciprocal lattice represents a set of planes.

3-D Reciprocal Lattice

Real space:Unit cell vectors: a,b,c

magnitude directiona d100 [100]b d010 [010]c d001 [001]

Reciprocal space:Unit cell vectors: a*,b*

magnitude directiona* 1/d100 ⊥ b and cb* 1/d010 ⊥ a and cc* 1/d001 ⊥ a and b

Note: as volume of unit cell in real space increases the volume of unit cell in reciprocal space decreases, and vice versa. a*,b* and c* are parallel to corresponding a,b and c, and this is only true for the unit cells of cubic, tetragonal and orthorhmbic crystal systems.

Orthorhombic

Convergent Beam Electron Diffraction (CBED)

CBED uses a conver-gent beam of elec-trons to limit area of specimen which con-tributes to diffractionpattern.Each spot in SAED thenbecomes a disc withinwhich variations inintensity can be seen.CBED patterns containa wealth of informationabout symmetry andthickness of specimen.Big advantage of CBEDis that the informationis generated from smallregions beyond reachof other techniques.

Why CBED?

Symmetry Deviations

Bright and Dark Field Imaging

BF

DF

DFMicrostructure of a Pb(ZrSnTi)O3specimen. → direction of structural modulation.

• • •

••

•

SAED vs CBED

spots disks

lens

sample

Parallel beam Convergent beam

T D T D

Convergence angle

objective

SAED CBED

Spatial resolution

>0.5µm

Spatial resolutionbeam size

Applications of CBED

•Phase identification•Symmetry determination-point and space group•Phase fingerprinting•Thickness measurement•Strain and lattice parameter measurement •Structure factor determination

Phase Identification in BaAl2Si2O8

200oC 400oC 800oC

<0001>

Hexagonal Orthorhombic Hexagonal

6mm 2mm 6mm

CBED

Symmetry and Lattice Parameter Determination

BFEDS

[143]

[100]

[111]

CBED

CBED-HOLZ

Experimental simulated

Latticeparameters

SAED

010

001

0.2µm

NbA

B

A

A

A

A

B

B

B

B

CBED

Phase Fingerprinting

[001] CBEDpatterns of anantiferroelectricPbZrO3 singlecrystal specimenat (a) 20oC, (b)280oC, (c)220oC.(d) [001] CBEDpattern of a rhombohedral ferroelectric Pb(ZrTi)O3Specimen at 20oC.

Orthorhombic AFE Cubic PE

Rhombohedral FE Rhombohedral FE CBED

High Resolution Z-contrast ImagingAtomic Ordering in Ba(Mg1/3Nb2/3)O3

(STEM)

aI Z2

∆f: -52 -64 -76 -88 -100nm

HREM-Phase contrast

Z-contrast

A: disordered B: ordered region

XRD, SEM and TEM Studies of Nanocrystalline BaTiO3 Thin Film

10nm

AES

XRDSEM SEM

SEM

TEM

TEM

TEM

T (OC)

K

K-dielectric constant

Why are there so many spots?The Ewald Sphere

Ewaldcircle

0

incidentbeam

diffractedbeam

2θ

C

kGg

130

H

At G, diffraction occurs, at H no diffraction

Ewald’s sphere is built for interpreting diffraction

CG-C0=0G or kd-ki=g Laue equation

Ewald’s Sphere

Construction of Ewald’s Sphere

Ewald’s sphere is built for interpreting diffraction patterns and it shows which sets of planes are at (or close to) their Bragg angle for diffraction to occur.

•Incident wave is represented by a reciprocal vector k (lkl=1/λ and points in the direction of wave).•Construct a circle with radius 1/λ (i.e., k), which passes through origin of reciprocal lattice, 0.•Wherever a reciprocal point touches the circle, Bragg’s law is obeyed and a diffracted beam will occur.•C0–incident beam and CG–diffracted beam. The angle between C0 and CG must be 2θ.•0G is the reciprocal vector g130 and has magnitude of 1/d130.

0G/2=lklsinθ, → 0G=2/λsinθ, → 1/d130=2/λsinθ130

→ λ=2d130sinθ130

Ewald’s Sphere Construction in 3D

In a single crystalonly a few sets ofplanes are orientedat their Braggangle at any onetime.

Lattice VectorsReal space lattice vectorcorresponds to directions in crystal and it can be defined as:

r=ua+vb+wca,b and c are unit cell vectors,u,v and w are components ofthe direction index [uvw].

A reciprocal lattice vectorcan be written as:g*=ha*+kb*+lc*

a*,b* and c* are reciprocal unit vectors, and h,k and l are the Miller indices of the plane (hkl).

Effect of Spacing of planes in Real Space on Length of Reciprocal Vector, g

In a crystal of any structure, ghkl is normal to the (hkl)plane and has a length inversely proportional to the interplanar spacing of the planes.

(111)-

d111-

[111]-

Streaking Reciprocal Lattice Points

Bragg’s law predicts diffraction at only precise Bragg angles for an infinite crystal. In TEM experiments, specimens are thin in at least onedimension (thickness).Effect of small dimensions is to allow diffraction over a range of angles close to Bragg angle, or as if reciprocal lattice points are stretched out in the thickness direction. The stretched reciprocal lattice points are called relrods.

Deviation Parameter, s

RelrodEwald sphere can intersectwith a relrod even when itmisses the actual reciprocallattice point. Diffraction, atreduced intensity, can stilloccur. Deviation parameter,s, defines how close a relrodIs to the Ewald sphere andDiffraction vector K is given:

K=g+s

sK

g

The s is defined to be positiveIn the direction of the beamAnd negative if it pointsUpwards (as here).

e-

Beam Intensity vs Deviation, s

Ewald’s Sphere and Diffraction Patterns

The curvature of the circle with respect to the reciprocal lattice, depends on the relative values of the wavelength λ, and the spacing of the lattice planes in the crystal, d.

Indexing Diffraction Pattern-ratio Technique

Any 2-D section of a reciprocal lattice can be defined by two vectors so only need to index 2 spots.

1.Choose one spot to be the origin andmeasure r1

2.measure the spacing of a second spot r2

3.measure the angle, φ4.prepare a table giving the ratios of the

spacings of permitted diffraction planesin the known structure

5.take measured ratio r1/r2 and locate a value close to this in the table

6.assign more widely-spaced plane (lower indices) to the shorter r value

7.calculate angle between pair of planes of the type you haveindexed

8.if measured φ agrees with one of possible value, accept indexing. if not, revisit the table and select another possible pair of planes

9.finish indexing the pattern by vector addition.

Indexing Electron Diffraction Patterns

If we know the index for two diffraction spotsIt is possible to index the rest of the spots byUsing vector addition as shown. Every spotsCan be reached by a combination of these twoVectors.

Kikuchi Lines-1 e-

In a thick enough specimen,inelastic scattering (in 3-D)also take place.

Inelastically scattered e-s travel in all directions but their distribution peaks in a forwarddirection.

More are scattered forward Than sideways. This contri-butes a grey background around the central spot of thediffraction pattern, as shown.

Kikuchi Lines-2

Excess Line

Deficient Line

Projection of hkl plane

1.Inelastically scattered e-s can bediffracted only if they are travelingat Bragg angle,θB to a set of planes.2.Two sets of e-s will be able to do this - those at +θB and those at -θB.

3.More e-s at A than B, onebright (excess) line and onedark (deficient) line result.4. e-s are scattered in all directions, diffracted e-s form a cone, not a beam resulted in Kikuchi lines.5.Spacing of pair of Kikuchilines is the same as spacingof diffracted spots from thesame plane.

Kikuchi Lines-3

Lens

Specimen

hkl planes

Excess line Deficient line

Grey background

Diffraction pattern

Kikuchi Line Patterns for Si

Kikuchi lines pass straight through transmitted and diffracted spots. Diffracting planes are tilted at exactly the Bragg angle to optic axis.

T

D

Crystal has been titled slightly away from Bragg angle, so that Kikuchi lines no longer pass through trans-mitted and diffracted spots.

T

Here the crystal is tilted so that more that one set of planes are diffracting. Each set of diffracting planes has its own pair of Kikuchi lines.

Indexing Diffraction Pattern-ratio technique

Any 2-D section of a reciprocal lattice can be defined by two vectors so only need to index 2 spots.

1.Choose one spot to be the origin andmeasure r1

2.measure the spacing of a second spot r2

3.measure the angle, φ4.prepare a table giving the ratios of the

spacings of permitted diffraction planesin the known structure

5.take measured ratio r1/r2 and locate a value close to this in the table

6.assign more widely-spaced plane (lower indices) to the shorter r value

7.calculate angle between pair of planes of the type you haveindexed

8.if measured φ agrees with one of possible value, accept indexing. if not, revisit the table and select another possible pair of planes

9.finish indexing the pattern by vector addition.

Indexing - Example1.Choose T as the origin, r1-7.75mm

2.r2-12.87mm

3.φ~72O

4.Get a table giving relative reciprocal lattice spacings

5.r2/r1=1.66 gives several possible pair of planes in the table

r1

r2 φ

T

r=Lλ/d r2/r1=d1/d2

6.From the table of interplanar angle in cubic, φ~72O gives only one matched pair of planes, {100} (or {200}) and {311} for a face-centered lattice. {100} diffraction is not allowed in a face-centered structure.

7.Calculating interplanar angles leads to (131) or (113) and (200) angle between (311) and (200) is 25.2O.

8.Zone axis of pattern: r1 x r2 = [013] for (131) and (200) pair

Cubic

200

131 (311)?

_

[013]_

Formation of HOLZ Lines